Product of Functions of Bounded Variation is of Bounded Variation

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Theorem

Let $a, b$ be real numbers with $a < b$.

Let $f, g : \closedint a b \to \R$ be functions of bounded variation.

Let $\map {V_f} {\closedint a b}$ and $\map {V_g} {\closedint a b}$ be the total variations of $f$ and $g$ respectively.


Then the pointwise product $f \cdot g$ is of bounded variation with:

$\map {V_{f \cdot g} } {\closedint a b} \le A \map {V_f} {\closedint a b} + B \map {V_g} {\closedint a b}$

where:

$\map {V_{f \cdot g} } {\closedint a b}$ denotes the total variation of $f \cdot g$ on $\closedint a b$
$A, B$ are non-negative real numbers.


Proof

For each finite subdivision $P$ of $\closedint a b$, write:

$P = \set {x_0, x_1, \ldots, x_n }$

with:

$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$

By Function of Bounded Variation is Bounded:

$f$ and $g$ are bounded.

So, there exists $A, B \in \R$ such that:

$\size {\map f x} \le B$
$\size {\map g x} \le A$

for all $x \in \closedint a b$.

Then:

\(\ds \map {V_{f \cdot g} } {P ; \closedint a b}\) \(=\) \(\ds \sum_{i \mathop = 1}^n \size {\map {\paren {f \cdot g} } {x_i} - \map {\paren {f \cdot g} } {x_{i - 1} } }\) using the notation from the definition of bounded variation
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \size {\map f {x_i} \map g {x_i} - \map f {x_{i - 1} } \map g {x_{i - 1} } }\)
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \size {\map f {x_i} \map g {x_i} - \map f {x_{i - 1} } \map g {x_i} + \map f {x_{i - 1} } \map g {x_i} - \map f {x_{i - 1} } \map g {x_{i - 1} } }\)
\(\ds \) \(\le\) \(\ds \sum_{i \mathop = 1}^n \size {\map f {x_i} \map g {x_i} - \map f {x_{i - 1} } \map g {x_i} } + \sum_{i \mathop = 1}^n \size {\map f {x_{i - 1} } \map g {x_i} - \map f {x_{i - 1} } \map g {x_{i - 1} } }\) Triangle Inequality
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \size {\map g {x_i} } \size {\map f {x_i} - \map f {x_{i - 1} } } + \sum_{i \mathop = 1}^n \size {\map f {x_{i - 1} } } \size {\map g {x_i} - \map g {x_{i - 1} } }\)
\(\ds \) \(\le\) \(\ds A \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } } + B \sum_{i \mathop = 1}^n \size {\map g {x_i} - \map g {x_{i - 1} } }\) since $\size {\map g {x_i} } \le A$ and $\size {\map f {x_{i - 1} } } \le B$
\(\ds \) \(=\) \(\ds A \map {V_f} {P ; \closedint a b} + B \map {V_g} {P ; \closedint a b}\)

Since $f$ and $g$ are of bounded variation, there exists $M, K \in \R$ such that:

$\map {V_f} {P ; \closedint a b} \le M$
$\map {V_g} {P ; \closedint a b} \le K$

for all finite subdivisions $P$.

We therefore have:

$\map {V_{f \cdot g} } {P ; \closedint a b} \le A M + B K$

so $f \cdot g$ is of bounded variation.

Further, we have:

\(\ds \map {V_{f \cdot g} } {\closedint a b}\) \(=\) \(\ds \sup_P \paren {\map {V_{f \cdot g} } {P ; \closedint a b} }\) Definition of Total Variation of Real Function on Closed Bounded Interval
\(\ds \) \(\le\) \(\ds \sup_P \paren {A \map {V_f} {P ; \closedint a b} } + \sup_P \paren {B \map {V_g} {P ; \closedint a b} }\)
\(\ds \) \(=\) \(\ds A \sup_P \paren {\map {V_f} {P ; \closedint a b} } + B \sup_P \paren {\map {V_g} {P ; \closedint a b} }\) Multiple of Supremum
\(\ds \) \(=\) \(\ds A \map {V_f} {\closedint a b} + B \map {V_g} {\closedint a b}\) Definition of Total Variation of Real Function on Closed Bounded Interval

$\blacksquare$


Sources

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